Defines mass for non-smooth hyperbolic spaces using a modified flow.
problem Defining mass for non-smooth, asymptotically hyperbolic spaces.
method Normalized Ricci-DeTurck flow with scalar curvature lower bound.
result Mass function well-defined for continuous metrics.
We show that the polyhomogeneity at infinity of an asymptotically complex hyperbolic metric is preserved along the Ricci-DeTurck flow. Moreover, if the initial metric is `smooth up to the boundary', this will be preserved by the Ricci-DeTurck flow and the normalized Ricci flow. When the initial metric is Kähler, sharpe…
Volume comparison theorem for rank 1 symmetric spaces proved.
problem Volume comparison for symmetric spaces of non-compact type.
method Normalized Ricci--DeTurck flow to analyze volume functional and derive monotonicity properties.
result Volume comparison theorem established for rank 1 symmetric spaces of non-compact type.
Study Ricci-Deturck flow from rough metrics, proving short-time existence.
problem Short-time existence of Ricci-Deturck flow from rough metrics.
method Ricci-Deturck flow, bi-Lipschitz metrics, small gradient concentration.
result Proved short-time existence of Ricci-Deturck flow.
Study short-time existence of Ricci-DeTurck flow from rough metrics with Morrey-type integrability.
problem Short-time existence of Ricci-DeTurck flow from rough metrics with specific integrability condition.
method Rough existence theory, preservation and improvement of scalar curvature bounds.
result Preservation and improvement of distributional scalar curvature lower bounds under certain conditions.
We prove a rigidity result for non-negative scalar curvature perturbations of the Euclidean metric on Rn , which may be regarded as a weak version of the rigidity statement of the positive mass theorem. We prove our result by analyzing long time solutions of Ricci DeTurck flow. As a byproduct in doing so, w…
Study compares volumes of hyperbolic 3-manifolds using Ricci-DeTurck flow.
problem Volume comparison on finite-volume hyperbolic 3-manifolds.
method Exponential convergence of Ricci-DeTurck flow to the hyperbolic metric.
result The hyperbolic metric minimizes volume among metrics with bounded scalar curvature.
Along the Ricci flow, we study the polyhomogeneity of complete Riemannian metrics endowed with "a Lie structure fibred at infinity", that is, a class of Lie structures at infinity that induce in a precise way a fibre bundle structure on a certain compactification by a manifold with corners. When the compactification is…
Here, we study the existence and uniqueness of solutions to the Ricci flow on Finsler surfaces and show short time existence of solutions for such flows. To this purpose, we first study the Finslerian Ricci-DeTurck flow on Finsler surfaces and find a unique short time solution to this flow. Then, we find a solution to …
Study shows a mass quantity for C0 metrics that agrees with ADM mass.
problem Understanding ADM mass for C0 metrics and its behavior under Ricci-DeTurck flow. method Developed a C0 mass quantity and analyzed its behavior under Ricci-DeTurck flow. result The C0 mass at infinity is independent of coordinate charts and has controlled distortion under Ricci-DeTurck flow. Smooths metrics with nonnegative scalar curvature near singular sets.
problem Approximating metrics with nonnegative scalar curvature near singularities.
method Ricci-DeTurck flow to approximate metrics.
result Approximated metrics converge to the original metric in C∞ away from the singular set. Ricci flow stabilizes hyperbolic 3-manifolds near the hyperbolic metric.
problem Stability of Ricci flow on hyperbolic 3-manifolds.
method Normalized Ricci-DeTurck flow with exponential convergence to the hyperbolic metric.
result Normalized Ricci-DeTurck flow converges exponentially to the hyperbolic metric.
The paper discusses a flow for almost continuous metrics with bounded curvature, leading to smooth metrics with bounded scalar curvature.
problem Riemannian manifolds with almost continuous metrics and bounded curvature.
method Ricci-DeTurck flow applied to (1−ε0(n))h≤g0≤(1+ε0(n))h result Smooth metrics with bounded scalar curvature can be obtained from almost continuous metrics.
Harmonic gauge simplifies geometric analysis of Riemannian metrics.
problem Analyzing the Hilbert-Einstein functional and its stability.
method Developed a harmonic gauge to eliminate divergence terms and induce elliptic structure.
result Positivity of curvature operator implies spectral stability of the functional.
The paper establishes bounds on scalar curvature on asymptotically flat manifolds.
problem Establishing scalar curvature bounds on asymptotically flat manifolds.
method Using Ricci-DeTurck flow and distributional scalar curvature, the paper derives bounds on scalar curvature.
result The scalar curvature lower bound under Ricci-DeTurck flow depends on the scalar curvature lower bound in the β-weak sense and time.
It is the purpose of this article to establish a technical tool to study regularity of solutions to parabolic equations on manifolds. As applications of this technique, we prove that solutions to the Ricci-DeTurck flow, the surface diffusion flow and the mean curvature flow enjoy joint analyticity in time and space, an…
Uniform proof for Ricci flows on complete manifolds.
problem Proving short-time existence, uniqueness, and continuous dependence for Ricci flows.
method Using Koch-Lamm framework and tensor heat kernel estimates, with a new continuous dependence estimate.
result Uniform proof of short-time existence, uniqueness, and continuous dependence for Ricci flows.
The paper constructs Ricci flow solutions for non-smooth metrics in four dimensions.
problem Constructing Ricci flow solutions for non-smooth metrics in four dimensions.
method Ricci DeTurck flow on closed manifolds with initial values in W2,2. result Constructs solutions to Ricci flow for non-smooth metrics in four dimensions.
We show the existence of a global unique and analytic solution for the mean curvature flow, the surface diffusion flow and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. We also show the existence of a global unique and analytic solution to the Ricci-DeTurck flow on euclidean s…
Recently, we have studied evolution of a family of Finsler metrics along Finsler Ricci flow and proved its convergence in short time. Here, existence of solutions to the so called Hamilton Ricci flow on Finsler spaces is studied and a short time solution is found. To this end the Finslerian Ricci-DeTurck flow on Finsle…
We study the short-time existence and regularity of solutions to a boundary value problem for the Ricci-DeTurck equation on a manifold with boundary. Using this, we prove the short-time existence and uniqueness of the Ricci flow prescribing the mean curvature and conformal class of the boundary, with arbitrary initial …
Simplified proof of stability for Ricci flow near ALE metrics.
problem Stability of Ricci flow near ALE metrics with integrable deformations.
method Equivalence between integrability and almost-orthogonality property of Ricci-DeTurck tensor, analysis in weighted Holder spaces.
result Dynamical stability of Ricci flow near linearly stable Ricci-flat ALE metrics.
Ricci flow stability on manifolds with bounded geometry ensures convergence to hyperbolic metrics.
problem Stability and convergence of Ricci flow on manifolds with bounded geometry.
method Continuous dependence on initial conditions, sectoriality of Ricci-DeTurck flow generator, and Hölder norm analysis.
result Ricci flow converges to hyperbolic metrics under certain conditions.
Higher-dimensional Ricci flows are shown to have unique and stable solutions.
problem Stability and uniqueness of Ricci flows in higher dimensions.
method Generalization of Bamler-Kleiner's proof to higher dimensions, use of Brendle's classification of κ-solutions, and maximum principle for linearized Ricci-DeTurck flow.
result Canonical evolution through singularities for manifolds with positive isotropic curvature.
We consider smooth, not necessarily complete, Ricci flows, (M,g(t))t∈(0,T) with Ric(g(t))≥−1 and ∣Rm(g(t))∣≤c/t for all t∈(0,T) coming out of metric spaces (M,d0) in the sense that (M,d(g(t)),x0)→(M,d0,x0) as t↘0 in the pointed Gromov-Hausdorff…
New method trains normalizing flows using entropy-regularized transport.
problem Training continuous normalizing flows efficiently.
method Formulates flows as gradients of scalar potentials, training only these potentials.
result Trains normalizing flows without explicit flow computation during training.
Variational inference relies on flexible approximate posterior distributions. Normalizing flows provide a general recipe to construct flexible variational posteriors. We introduce Sylvester normalizing flows, which can be seen as a generalization of planar flows. Sylvester normalizing flows remove the well-known single…
PL-MCMC samples from normalizing flows' conditional distributions.
problem Sampling from complex conditional distributions learned by normalizing flows.
method Metropolis-Hastings implementation of PL-MCMC.
result PL-MCMC asymptotically samples from exact conditional distributions.
Fractal Flow enhances normalizing flows with interpretable latent space and hierarchical modeling.
problem High-dimensional density estimation and generative modeling challenges.
method Integrates topic modeling (LDA) and fractal strategy into normalizing flows.
result Achieves latent clustering, controllable generation, and superior estimation accuracy.
SurVAE Flows combine VAEs and flows using surjective transformations.
problem Combining the strengths of VAEs and flows to model complex densities.
method Modular framework of composable deterministic and stochastic transformations.
result Exact likelihood computation and lower bound on likelihood.
Study shows limits of certain normalizing flows in higher dimensions.
problem Understanding the representation power of normalizing flows in different dimensions.
method Rigorously established bounds on expressive power of basic normalizing flows.
result Limited representation power in higher dimensions, especially with moderate depth.
The paper connects complex normalizing flows to Kähler-Ricci flows using geometric and statistical perspectives.
problem Understanding the relationship between complex normalizing flows and Kähler-Ricci flows.
method Develops connections between complex normalizing flows and Kähler-Ricci flows by relating the log determinant to Ricci curvature and using a Bayesian perspective.
result Reconciles the complex normalizing flow and Kähler-Ricci flow, showing they are related under certain conditions.
Riemannian flows model curved spaces better.
problem Normalizing flows misspecified on curved spaces.
method Riemannian continuous normalizing flows using ODE solutions.
result Improves modeling on spheres, torii, and hyperbolic spaces.
New framework explains normalizing flows' power and limitations.
problem Understanding the expressive power and limitations of normalizing flows.
method Theoretical framework for well-conditioned coupling-based normalizing flows and volume-preserving flows.
result RealNVP is distributionally universal, but volume-preserving flows are not.
A new method learns latent space normalizing flow for approximate inference in generator models.
problem Approximate inference in generator models with complex posterior distributions.
method Jointly learns latent space normalizing flow and generator model using MCMC-based maximum likelihood.
result The short-run Langevin flow approximates the posterior and aligns with the normalizing flow prior.
TCNF models SDEs using time deformation of Brownian motion.
problem Modeling SDEs with existing methods.
method Time-changed normalizing flows (TCNF) based on time deformation of Brownian motion.
result Improved modeling of SDEs, including Ornstein-Uhlenbeck process.
Paper introduces Categorical Normalizing Flows for better handling of categorical data.
problem Limited application of normalizing flows on categorical data due to lack of intrinsic order.
method Categorical Normalizing Flows use continuous transformations to model latent relations in categorical data, optimizing both continuous representation and model likelihood.
result GraphCNF, a permutation-invariant generative model, outperforms state-of-the-art on molecule generation.
This study evaluates different normalizing flow architectures for MCMC.
problem Lack of systematic comparison of normalizing flow architectures in MCMC.
method Extensive evaluation of various normalizing flow architectures on different MCMC methods and target distributions.
result Contractive residual flows are the best general-purpose models for MCMC.
Improved lattice field theory simulations with local-Autoregressive Conditional Normalizing Flow.
problem Efficiently sampling lattice field theories with computational challenges.
method Integrates locality into autoregressive conditional normalizing flows.
result Autocorrelation times improved by orders of magnitude for φ4 theory on a 2D lattice. Guillarmou extends X-ray transform to magnetic and thermostat flows.
problem Stability of magnetic X-ray transforms.
method Generalizes normal operator to thermostat and magnetic flows, proving ellipticity.
result Elliptic pseudodifferential operators of order -1 for generalized normal operators.
Improves normalizing flows by incorporating data dependencies.
problem Current normalizing flow learning assumes independent data, leading to errors.
method Proposes a likelihood objective with dependencies and efficient learning algorithm.
result Improves density estimation and data generation on real-world data.
New normalizing flows in hyperbolic space improve posterior modeling for hierarchical data.
problem Limited flexibility of existing normalizing flows in Euclidean space for hierarchical data.
method Elevated normalizing flows to hyperbolic spaces using coupling transforms and Wrapped Hyperboloid Coupling.
result Improved performance on density estimation and hierarchical graph data.
Normalizing flows are shown to be equivalent to Bayesian networks, revealing new insights.
problem Understanding the limitations and capabilities of normalizing flows.
method Revisiting normalizing flows as probabilistic graphical models and analyzing their structure.
result Normalizing flows can be reduced to Bayesian networks, revealing new insights into their structure and capabilities.
Self Normalizing Flows improve normalizing flows by reducing computational complexity.
problem Efficient gradient computation in normalizing flows, especially in Jacobian determinant terms.
method Introducing Self Normalizing Flows that replace expensive terms with learned approximate inverses.
result Models can be trained more quickly and perform better than functionally constrained counterparts.
TTF improves performance of normalizing flows for heavy-tailed distributions.
problem Improving performance of normalizing flows for heavy-tailed distributions.
method Uses a Gaussian base distribution and a final transformation layer to produce heavy tails.
result Experimental results show TTF outperforms current methods, especially in high-dimensional or heavy-tailed scenarios.
The study examines the limitations of bi-Lipschitz Normalizing Flows in approximating certain distributions.
problem The expressivity of bi-Lipschitz Normalizing Flows in approximating specific target distributions.
method Characterization of expressivity through lower bounds on Total Variation distance and discussion of potential remedies.
result Several target distributions are difficult to approximate using bi-Lipschitz Normalizing Flows, and lower bounds on their approximation are provided.
ImpFlows generalize normalizing flows by implicitly defining transformations.
problem Creating flexible and tractable probability distributions.
method Implicitly defined invertible transformations using roots of equations.
result ImpFlows can represent functions that ResFlows cannot, with comparable parameters.
New proof shows coupling-based flows converge linearly to diagonalize data covariance.
problem Understanding convergence of coupling-based normalizing flows to arbitrary data distributions.
method Proved linear convergence rate for whitening of data distribution.
result Coupling-based flows achieve linear convergence to diagonalize data covariance.